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A data-dependent skeleton estimate and a scale-sensitive dimension for classification

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Abstract

The classical binary classification problem is investigated when it is known in advance that the posterior probability function (or regression function) belongs to some class of functions. We introduce and analyze a method which effectively exploits this knowledge. The method is based on minimizing the empirical risk over a carefully selected ``skeleton'' of the class of regression functions. The skeleton is a covering of the class based on a data--dependent metric, especially fitted for classification. A new scale--sensitive dimension is introduced which is more useful for the studied classification problem than other, previously defined, dimension measures. This fact is demonstrated by performance bounds for the skeleton estimate in terms of the new dimension.

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  • Marta Horvath & Gábor Lugosi, 1996. "A data-dependent skeleton estimate and a scale-sensitive dimension for classification," Economics Working Papers 199, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:199
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    1. Gábor Lugosi & Andrew B. Nobel, 1998. "Adaptive model selection using empirical complexities," Economics Working Papers 323, Department of Economics and Business, Universitat Pompeu Fabra.
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    Cited by:

    1. Devroye, Luc & Györfi, Laszlo & Krzyzak, Adam, 1998. "The Hilbert Kernel Regression Estimate," Journal of Multivariate Analysis, Elsevier, vol. 65(2), pages 209-227, May.
    2. Kohler, Michael & Máthé, Kinga & Pintér, Márta, 2002. "Prediction from Randomly Right Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 73-100, January.
    3. Boumaza, Rachid, 2004. "Discriminant analysis with independently repeated multivariate measurements: an L2 approach," Computational Statistics & Data Analysis, Elsevier, vol. 47(4), pages 823-843, November.
    4. Fischer, Aurélie, 2010. "Quantization and clustering with Bregman divergences," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2207-2221, October.
    5. Mojirsheibani, Majid, 2001. "An iterated classification rule based on auxiliary pseudo-predictors," Computational Statistics & Data Analysis, Elsevier, vol. 38(2), pages 125-138, December.
    6. Biau, Gérard & Devroye, Luc, 2005. "Density estimation by the penalized combinatorial method," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 196-208, May.
    7. Mojirsheibani, Majid, 2002. "An Almost Surely Optimal Combined Classification Rule," Journal of Multivariate Analysis, Elsevier, vol. 81(1), pages 28-46, April.
    8. Kohler, Michael, 1999. "Universally Consistent Regression Function Estimation Using Hierarchial B-Splines," Journal of Multivariate Analysis, Elsevier, vol. 68(1), pages 138-164, January.
    9. Camerlenghi, F. & Capasso, V. & Villa, E., 2014. "On the estimation of the mean density of random closed sets," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 65-88.

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    More about this item

    Keywords

    Estimation; hypothesis testing; statistical decision theory: operations research;
    All these keywords.

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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